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Question Number 72894 by TawaTawa last updated on 04/Nov/19

Answered by mind is power last updated on 04/Nov/19

$$\mathrm{lets}\mathrm{assum}\mathrm{n},\mathrm{m}\mathrm{are}\mathrm{relativ}\mathrm{prime}$$$$$$$$\mathrm{let}\mathrm{N}={\mathrm{p}}_1^{\mathrm{k}1}.{\mathrm{p}}_2^{\mathrm{k}2}....{\mathrm{p}}_{\mathrm{n}}^{\mathrm{kn}}$$$$\sigma \left(\mathrm{N}\right)=\underset{{\mathrm{l}}_1=0}{\sum ^{{\mathrm{k}}_1}}\underset{\mathrm{l}2=0}{\sum ^{\mathrm{k}2}}....\underset{\ln =0}{\sum ^{\mathrm{kn}}}{\stackrel{\mathrm{l}1}{\mathrm{p}}}_1{\stackrel{\mathrm{l}2}{\mathrm{p}}}_2...{\stackrel{\ln }{\mathrm{p}}}_{\mathrm{n}}=\underset{\mathrm{i}=1}{\prod ^{\mathrm{n}}}\left(\underset{\mathrm{li}=0}{\sum ^{\mathrm{ki}}}{\mathrm{p}}_{\mathrm{i}}^{\mathrm{li}}\right)$$$$\sigma \left(\mathrm{N}\right)=\underset{\mathrm{i}=1}{\prod ^{\mathrm{n}}}\frac{{\mathrm{p}}_{\mathrm{i}}^{\mathrm{ki}+1}-1}{{\mathrm{p}}_{\mathrm{i}}-1}$$$${\mathrm{N}}_3,\sigma \left({\mathrm{N}}_3\right)⩾3.{\mathrm{N}}_3$$$$\Rightarrow \underset{\mathrm{i}=1}{\prod ^{\mathrm{n}}}\frac{{\mathrm{p}}_{\mathrm{i}}^{\mathrm{ki}+1}-1}{\mathrm{pi}-1}⩾3.{\mathrm{p}}_1^{\mathrm{k}1}....{\mathrm{p}}_{\mathrm{n}}^{\mathrm{kn}}$$$$\iff (1+{\mathrm{p}}_{\mathrm{i}}....+{\mathrm{p}}_{\mathrm{i}}^{\mathrm{k}1})........(1+{\mathrm{p}}_{\mathrm{n}}+....+{\mathrm{p}}_{\mathrm{n}}^{\mathrm{kn}})⩾{3\mathrm{p}}_1^{\mathrm{k}1}...{\mathrm{p}}_{\mathrm{n}}^{\mathrm{kn}}$$$$\sigma (3^2{5}^2{2}^2=900)=\frac{26}{2}.\frac{124}{4}.\frac{7}{1}=13.31.7=2821⩾2700$$$${\mathrm{N}}_3=900$$$$\sigma \left(2^3{3}^4{5}^3\right)=400140⩾4.2^3{3}^4{5}^3$$$${\mathrm{N}}_4=81000$$

Commented by TawaTawa last updated on 04/Nov/19

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}$$

Commented by mind is power last updated on 04/Nov/19

$${\mathrm{y}}^{\prime }\mathrm{re}\mathrm{welcom}$$$$$$

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